Problem: Multiply the following complex numbers: $({1+4i}) \cdot ({1-i})$
Answer: Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({1+4i}) \cdot ({1-i}) = $ $ ({1} \cdot {1}) + ({1} \cdot {-1}i) + ({4}i \cdot {1}) + ({4}i \cdot {-1}i) $ Then simplify the terms: $ (1) + (-1i) + (4i) + (-4 \cdot i^2) $ Imaginary unit multiples can be grouped together. $ 1 + (-1 + 4)i - 4i^2 $ After we plug in $i^2 = -1$ , the result becomes $ 1 + (-1 + 4)i - (-4) $ The result is simplified: $ (1 + 4) + (3i) = 5+3i $